The upper domatic number of a graph

Abstract

Let G=(V,E) be a graph. For two disjoint sets of vertices R and S, set R dominates set S if every vertex in S is adjacent to at least one vertex in R. In this paper we introduce the upper domatic numberD(G), which equals the maximum order k of a vertex partition π={V1,V2,…,Vk} such that for every i,j, 1≤i<j≤k, either Vi dominates Vj or Vj dominates Vi, or both. We study properties of the upper domatic number of a graph, determine bounds on D(G), and compare D(G) to a related parameter, the transitivity Tr(G) of G.